Distribution-Free Bayesian multivariate predictive inference

We introduce a comprehensive Bayesian multivariate predictive inference framework. The basis for our framework is a hierarchical Bayesian model, that is a mixture of finite Polya trees corresponding to multiple dyadic partitions of the unit cube. Given a sample of observations from an unknown multivariate distribution, the posterior predictive distribution is used to model and generate future observations from the unknown distribution. We illustrate the implementation of our methodology and study its performance on simulated examples. We introduce an algorithm for constructing conformal prediction sets, that provide finite sample probability assurances for future observations, with our Bayesian model.